Collection Editor:
Free High School Science Texts Project
Collection Editor:
Free High School Science Texts Project Authors:
Free High School Science Texts Project Rory Adams
Mark Horner Heather Williams
Online:
< http://cnx.org/content/col11239/1.2/ >
C O N N E X I O N S
Rice University, Houston, Texas
Collection structure revised: August 5, 2011 PDF generated: October 29, 2012
For copyright and attribution information for the modules contained in this collection, see p. 265.
Review of Past Work . . . 1
Rational Numbers . . . .. . . 19
Exponentials . . . 27
Estimating Surds . . . 35
Irrational Numbers and Rounding O . . . 39
1 Number patterns 1.1 Introduction . . . .. . . 43
1.2 Notation . . . 45
Solutions . . . 49
2 Finance 2.1 Foreign exchange rates . . . 51
2.2 Simple interest . . . 56
2.3 Compound interest . . . 59
Solutions . . . 65
3 Products and Factors . . . 69
4 Equations and inequalities 4.1 Solving linear equations . . . 83
4.2 Solving quadratic equations . . . 86
4.3 Exponential equations . . . 89
4.4 Linear inequalities . . . 92
4.5 Linear simultaneous equations . . . 94
4.6 Mathematical models . . . .. . . 97
Solutions . . . 100
5 Functions and graphs 5.1 Introduction and key concepts . . . .. . . 109
5.2 The straight line . . . 117
5.3 The parabola . . . .. . . 120
5.4 Hyperbolic functions . . . 126
5.5 Exponential functions . . . 130
Solutions . . . 136
6 Average gradient 6.1 Straight line functions . . . 137
6.2 Parabolic functions . . . .. . . 138
Solutions . . . 140
7 Geometry basics 7.1 Points, lines and angles . . . .. . . 141
7.2 Triangles . . . 150
7.3 Quadrilaterals . . . 156
7.4 Polygons . . . 158
8 Geometry 8.1 Right prisms and cylinders . . . 165
8.2 Polygons . . . 169
8.3 Analytical geometry . . . 171
8.4 Transformations . . . .. . . 176
Solutions . . . 187
9 Trigonometry 9.1 Introduction and key concepts . . . .. . . 191
9.2 The trig functions . . . 193
9.3 Applications . . . .. . . 197
9.4 Graphs of trig functions . . . 199
Solutions . . . 212
10 Statistics 10.1 Recap and sample data sets . . . 215
10.2 Graphical representation of data . . . 223
10.3 Summarising data . . . .. . . 228
10.4 Misuse of statistics . . . .. . . 233
Solutions . . . 240
11 Probability 11.1 Random experiments . . . .. . . 245
11.2 Key concepts . . . .. . . 249
Solutions . . . 258
Glossary . . . 261
Index . . . 263
Attributions . . . .265
Review of Past Work
Introduction
This chapter describes some basic concepts which you have seen in earlier grades and lays the foundation for the remainder of this book. You should feel condent with the content in this chapter, before moving on with the rest of the book.
You can try out your skills on exercises in this chapter and ask your teacher for more questions just like them. You can also try to make up your own questions, solve them and try them out on your classmates to see if you get the same answers.
Practice is the only way to get good at maths!
What is a number?
A number is a way to represent quantity. Numbers are not something that you can touch or hold, because they are not physical. But you can touch three apples, three pencils, three books. You can never just touch three, you can only touch three of something. However, you do not need to see three apples in front of you to know that if you take one apple away, there will be two apples left. You can just think about it. That is your brain representing the apples in numbers and then performing arithmetic on them.
A number represents quantity because we can look at the world around us and quantify it using numbers.
How many minutes? How many kilometers? How many apples? How much money? How much medicine?
These are all questions which can only be answered using numbers to tell us how much of something we want to measure.
A number can be written in many dierent ways and it is always best to choose the most appropriate way of writing the number. For example, a half may be spoken aloud or written in words, but that makes mathematics very dicult and also means that only people who speak the same language as you can understand what you mean. A better way of writing a half is as a fraction 12 or as a decimal number0,5. It is still the same number, no matter which way you write it.
In high school, all the numbers which you will see are called real numbers and mathematicians use the symbolRto represent the set of all real numbers, which simply means all of the real numbers. Some of these real numbers can be written in ways that others cannot. Dierent types of numbers are described in detail in Section 1.12.
Sets
A set is a group of objects with a well-dened criterion for membership. For example, the criterion for belonging to a set of apples, is that the object must be an apple. The set of apples can then be divided into red apples and green apples, but they are all still apples. All the red apples form another set which is a sub-set of the set of apples. A sub-set is part of a set. All the green apples form another sub-set.
1This content is available online at <http://cnx.org/content/m31330/1.4/>.
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Now we come to the idea of a union, which is used to combine things. The symbol for union is ∪. Here, we use it to combine two or more intervals. For example, ifxis a real number such that1< x≤3 or 6≤x <10 , then the set of all the possiblexvalues is:
(1,3]∪[6,10) (1)
where the∪sign means the union (or combination) of the two intervals. We use the set and interval notation and the symbols described because it is easier than having to write everything out in words.
Letters and Arithmetic
The simplest thing that can be done with numbers is adding, subtracting, multiplying or dividing them.
When two numbers are added, subtracted, multiplied or divided, you are performing arithmetic2. These four basic operations can be performed on any two real numbers.
Mathematics as a language uses special notation to write things down. So instead of:
oneplusoneisequaltotwo (2)
mathematicians write
1 + 1 = 2 (3)
In earlier grades, place holders were used to indicate missing numbers in an equation.
1 += 2 4−= 2 + 3−2= 2
(4)
However, place holders only work well for simple equations. For more advanced mathematical workings, letters are usually used to represent numbers.
1 +x= 2 4−y= 2 z+ 3−2z= 2
(5)
These letters are referred to as variables, since they can take on any value depending on what is required.
For example,x= 1in (5), butx= 26 in 2 +x= 28.
A constant has a xed value. The number 1 is a constant. The speed of light in a vacuum is also a constant which has been dened to be exactly 299 792 458 m·s−1(read metres per second). The speed of light is a big number and it takes up space to always write down the entire number. Therefore, letters are also used to represent some constants. In the case of the speed of light, it is accepted that the letterc represents the speed of light. Such constants represented by letters occur most often in physics and chemistry.
Additionally, letters can be used to describe a situation mathematically. For example, the following equation
x+y=z (6)
can be used to describe the situation of nding how much change can be expected for buying an item. In this equation, y represents the price of the item you are buying, x represents the amount of change you
2Arithmetic is derived from the Greek word arithmos meaning number.
should get back andzis the amount of money given to the cashier. So, if the price is R10 and you gave the cashier R15, then write R15 instead ofzand R10 instead ofy and the change is thenx.
x+ 10 = 15 (7)
We will learn how to solve this equation towards the end of this chapter.
Addition and Subtraction
Addition (+) and subtraction (−) are the most basic operations between numbers but they are very closely related to each other. You can think of subtracting as being the opposite of adding since adding a number and then subtracting the same number will not change what you started with. For example, if we start with aand addb, then subtract b, we will just get back toaagain:
a+b−b=a
5 + 2−2 = 5 (8)
If we look at a number line, then addition means that we move to the right and subtraction means that we move to the left.
The order in which numbers are added does not matter, but the order in which numbers are subtracted does matter. This means that:
a+b = b+a
a−b 6= b−a ifa6=b (9)
The sign6=means is not equal to. For example,2 + 3 = 5 and3 + 2 = 5, but5−3 = 2 and3−5 =−2.
−2is a negative number, which is explained in detail in "Negative Numbers" (Section : Negative Numbers).
Commutativity for Addition
The fact thata+b=b+a, is known as the commutative property for addition.
Multiplication and Division
Just like addition and subtraction, multiplication (×, ·) and division (÷, /) are opposites of each other.
Multiplying by a number and then dividing by the same number gets us back to the start again:
a×b÷b=a
5×4÷4 = 5 (10)
Sometimes you will see a multiplication of letters as a dot or without any symbol. Don't worry, its exactly the same thing. Mathematicians are ecient and like to write things in the shortest, neatest way possible.
abc = a×b×c
a·b·c = a×b×c (11)
It is usually neater to write known numbers to the left, and letters to the right. So although 4xand x4 are the same thing, it looks better to write 4x. In this case, the 4 is a constant that is referred to as the coecient ofx.
Commutativity for Multiplication
The fact that ab = ba is known as the commutative property of multiplication. Therefore, both addition and multiplication are described as commutative operations.
Brackets
Brackets3in mathematics are used to show the order in which you must do things. This is important as you can get dierent answers depending on the order in which you do things. For example:
(5×5) + 20 = 45 (12)
whereas
5×(5 + 20) = 125 (13)
If there are no brackets, you should always do multiplications and divisions rst and then additions and subtractions4. You can always put your own brackets into equations using this rule to make things easier for yourself, for example:
a×b+c÷d = (a×b) + (c÷d)
5×5 + 20÷4 = (5×5) + (20÷4) (14)
If you see a multiplication outside a bracket like this a(b+c)
3 (4−3) (15)
then it means you have to multiply each part inside the bracket by the number outside a(b+c) = ab+ac
3 (4−3) = 3×4−3×3 = 12−9 = 3 (16) unless you can simplify everything inside the bracket into a single term. In fact, in the above example, it would have been smarter to have done this
3 (4−3) = 3×(1) = 3 (17)
It can happen with letters too
3 (4a−3a) = 3×(a) = 3a (18)
3Sometimes people say parentheses instead of brackets.
4Multiplying and dividing can be performed in any order as it doesn't matter. Likewise it doesn't matter which order you do addition and subtraction. Just as long as you do any×÷before any+−.
Distributivity
The fact thata(b+c) =ab+ac is known as the distributive property.
If there are two brackets multiplied by each other, then you can do it one step at a time:
(a+b) (c+d) = a(c+d) +b(c+d)
= ac+ad+bc+bd (a+ 3) (4 +d) = a(4 +d) + 3 (4 +d)
= 4a+ad+ 12 + 3d
(19)
Negative Numbers
What is a negative number?
Negative numbers can be very confusing to begin with, but there is nothing to be afraid of. The numbers that are used most often are greater than zero. These numbers are known as positive numbers.
A negative number is a number that is less than zero. So, if we were to take a positive number a and subtract it from zero, the answer would be the negative ofa.
0−a=−a (20)
On a number line, a negative number appears to the left of zero and a positive number appears to the right of zero.
Figure 1: On the number line, numbers increase towards the right and decrease towards the left.
Positive numbers appear to the right of zero and negative numbers appear to the left of zero.
Working with Negative Numbers
When you are adding a negative number, it is the same as subtracting that number if it were positive.
Likewise, if you subtract a negative number, it is the same as adding the number if it were positive. Numbers are either positive or negative and we call this their sign. A positive number has a positive sign (+) and a negative number has a negative sign (−).
Subtraction is actually the same as adding a negative number.
In this example, aandbare positive numbers, but−bis a negative number a−b=a+ (−b)
5−3 = 5 + (−3) (21)
So, this means that subtraction is simply a short-cut for adding a negative number and instead of writing a+ (−b), we writea−b. This also means that−b+ais the same asa−b. Now, which do you nd easier to work out?
Most people nd that the rst way is a bit more dicult to work out than the second way. For example, most people nd 12−3 a lot easier to work out than −3 + 12, even though they are the same thing. So a−b, which looks neater and requires less writing is the accepted way of writing subtractions.
Table 1 shows how to calculate the sign of the answer when you multiply two numbers together. The rst column shows the sign of the rst number, the second column gives the sign of the second number and the third column shows what sign the answer will be.
a b a×bor a÷b
+ + +
+ − −
− + −
− − +
Table 1: Table of signs for multiplying or dividing two numbers.
So multiplying or dividing a negative number by a positive number always gives you a negative number, whereas multiplying or dividing numbers which have the same sign always gives a positive number. For example,2×3 = 6and−2× −3 = 6, but−2×3 =−6 and2× −3 =−6.
Adding numbers works slightly dierently (see Table 2). The rst column shows the sign of the rst number, the second column gives the sign of the second number and the third column shows what sign the answer will be.
a b a+b
+ + +
+ − ?
− + ?
− − −
Table 2: Table of signs for adding two numbers.
If you add two positive numbers you will always get a positive number, but if you add two negative numbers you will always get a negative number. If the numbers have a dierent sign, then the sign of the answer depends on which one is bigger.
Living Without the Number Line
The number line in Figure 1 is a good way to visualise what negative numbers are, but it can get very inecient to use it every time you want to add or subtract negative numbers. To keep things simple, we will write down three tips that you can use to make working with negative numbers a little bit easier. These tips will let you work out what the answer is when you add or subtract numbers which may be negative, and will also help you keep your work tidy and easier to understand.
Negative Numbers Tip 1
If you are given an expression like−a+b, then it is easier to move the numbers around so that the expression looks easier. In this case, we have seen that adding a negative number to a positive number is the same as subtracting the number from the positive number. So,
−a+b = b−a
−5 + 10 = 10 + (−5)
= 10−5
= 5
(22)
This makes the expression easier to understand. For example, a question like What is−7 + 11? looks a lot more complicated than What is11−7?, even though they are exactly the same question.
Negative Numbers Tip 2
When you have two negative numbers like−3−7, you can calculate the answer by simply adding together the numbers as if they were positive and then putting a negative sign in front.
−c−d = −(c+d)
−7−2 = −(7 + 2) =−9 (23)
Negative Numbers Tip 3
In Table 2 we saw that the sign of two numbers added together depends on which one is bigger. This tip tells us that all we need to do is take the smaller number away from the larger one and remember to give the answer the sign of the larger number. In this equation,F is bigger thane.
e−F = −(F−e)
2−11 = −(11−2) =−9 (24)
You can even combine these tips: for example, you can use Tip 1 on−10 + 3 to get3−10 and then use Tip 3 to get−(10−3) =−7.
Negative Numbers 1. Calculate:
(a)(−5)−(−3) (b)(−4) + 2 (c) (−10)÷(−2)
(d)11−(−9) (e)−16−(6) (f) −9÷3×2
(g)(−1)×24÷8×(−3) (h)(−2) + (−7) (i)1−12 (j)3−64 + 1 (k)−5−5−5 (l)−6 + 25 (m)−9 + 8−7 + 6−5 + 4−3 + 2−1
Table 3 Click here for the solution5
5See the le at <http://cnx.org/content/m31330/latest/http://www.fhsst.org/l3B>
2. Say whether the sign of the answer is +or −
(a)−5 + 6 (b)−5 + 1 (c) −5÷ −5 (d)−5÷5 (e)5÷ −5 (f) 5÷5 (g)−5× −5 (h)−5×5 (i)5× −5 (j)5×5
Table 4 Click here for the solution6
Rearranging Equations
Now that we have described the basic rules of negative and positive numbers and what to do when you add, subtract, multiply and divide them, we are ready to tackle some real mathematics problems!
Earlier in this chapter, we wrote a general equation for calculating how much change (x) we can expect if we know how much an item costs (y) and how much we have given the cashier (z). The equation is:
x+y=z (25)
So, if the price is R10 and you gave the cashier R15, then write R15 instead ofz and R10 instead ofy.
x+ 10 = 15 (26)
Now that we have written this equation down, how exactly do we go about nding what the change is?
In mathematical terms, this is known as solving an equation for an unknown (xin this case). We want to re-arrange the terms in the equation, so that onlyxis on the left hand side of the =sign and everything else is on the right.
The most important thing to remember is that an equation is like a set of weighing scales. In order to keep the scales balanced, whatever is done to one side must be done to the other.
6See the le at <http://cnx.org/content/m31330/latest/http://www.fhsst.org/l3K>
Figure 2: An equation is like a set of weighing scales. In order to keep the scales balanced, you must do the same thing to both sides. So, if you add, subtract, multiply or divide the one side, you must add, subtract, multiply or divide the other side too.
Method: Rearranging Equations
You can add, subtract, multiply or divide both sides of an equation by any number you want, as long as you always do it to both sides.
So for our example we could subtracty from both sides
x+y = z
x+y−y = z−y x = z−y x = 15−10
= 5
(27)
Now we can see that the change is the price subtracted from the amount paid to the cashier. In the example, the change should be R5. In real life we can do this in our heads; the human brain is very smart and can do arithmetic without even knowing it.
When you subtract a number from both sides of an equation, it looks like you just moved a positive number from one side and it became a negative on the other, which is exactly what happened. Likewise, if you move a multiplied number from one side to the other, it looks like it changed to a divide. This is because you really just divided both sides by that number and a number divided by itself is just 1
a(5 +c) = 3a a(5 +c)÷a = 3a÷a
a
a ×(5 +c) = 3×aa 1×(5 +c) = 3×1
5 +c = 3
c = 3−5 =−2
(28)
However, you must be careful when doing this, as it is easy to make mistakes.
The following is the WRONG thing to do
5a+c = 3a
5 +c = 3 (29)
Can you see why it is wrong? It is wrong because we did not divide thec term by aas well. The correct thing to do is
5a+c = 3a
5 +c÷a = 3 c÷a = 3−5 =−2
(30)
Rearranging Equations
1. If3 (2r−5) = 27, then2r−5 =... Click here for the solution7
2. Find the value forxif0,5 (x−8) = 0,2x+ 11 Click here for the solution8 3. Solve9−2n= 3 (n+ 2) Click here for the solution9
4. Change the formulaP =A+Akt toA= Click here for the solution10 5. Solve forx: ax1 +bx1 = 1 Click here for the solution11
7See the le at <http://cnx.org/content/m31330/latest/http://www.fhsst.org/l3k>
8See the le at <http://cnx.org/content/m31330/latest/http://www.fhsst.org/l30>
9See the le at <http://cnx.org/content/m31330/latest/http://www.fhsst.org/l38>
10See the le at <http://cnx.org/content/m31330/latest/http://www.fhsst.org/l39>
11See the le at <http://cnx.org/content/m31330/latest/http://www.fhsst.org/l3X>
Fractions and Decimal Numbers
A fraction is one number divided by another number. There are several ways to write a number divided by another one, such asa÷b,a/band ab. The rst way of writing a fraction is very hard to work with, so we will use only the other two. We call the number on the top (left) the numerator and the number on the bottom (right) the denominator. For example, in the fraction1/5 or 15, the numerator is 1 and the denominator is 5.
Denition - Fraction
The word fraction means part of a whole.
The reciprocal of a fraction is the fraction turned upside down, in other words the numerator becomes the denominator and the denominator becomes the numerator. So, the reciprocal of 23 is 32.
A fraction multiplied by its reciprocal is always equal to 1 and can be written a
b × b
a = 1 (31)
This is because dividing by a number is the same as multiplying by its reciprocal.
Denition - Multiplicative Inverse
The reciprocal of a number is also known as the multiplicative inverse.
A decimal number is a number which has an integer part and a fractional part. The integer and the fractional parts are separated by a decimal point, which is written as a comma in South African schools. For example the number310014 can be written much more neatly as3,14.
All real numbers can be written as a decimal number. However, some numbers would take a huge amount of paper (and ink) to write out in full! Some decimal numbers will have a number which will repeat itself, such as0,33333...where there are an innite number of 3's. We can write this decimal value by using a dot above the repeating number, so 0,˙3 = 0,33333.... If there are two repeating numbers such as 0,121212...
then you can place dots12 on each of the repeated numbers0,˙1 ˙2 = 0,121212.... These kinds of repeating decimals are called recurring decimals.
Table 5 lists some common fractions and their decimal forms.
Fraction Decimal Form
1
20 0,05
1
16 0,0625
1
10 0,1
1
8 0,125
1
6 0,16 ˙6
1
5 0,2
1
2 0,5
3
4 0,75
Table 5: Some common fractions and their equivalent decimal forms.
12or a bar, like0,12
Scientic Notation
In science one often needs to work with very large or very small numbers. These can be written more easily in scientic notation, which has the general form
a×10m (32)
where ais a decimal number between 0 and 10 that is rounded o to a few decimal places. The m is an integer and if it is positive it represents how many zeros should appear to the right ofa. If mis negative, then it represents how many times the decimal place inashould be moved to the left. For example3,2×103 represents 32 000 and3,2×10−3 represents0,0032.
If a number must be converted into scientic notation, we need to work out how many times the number must be multiplied or divided by 10 to make it into a number between 1 and 10 (i.e. we need to work out the value of the exponentm) and what this number is (the value ofa). We do this by counting the number of decimal places the decimal point must move.
For example, write the speed of light which is 299 792 458 ms−1 in scientic notation, to two decimal places. First, determine where the decimal point must go for two decimal places (to nda) and then count how many places there are after the decimal point to determinem.
In this example, the decimal point must go after the rst 2, but since the number after the 9 is a 7, a= 3,00.
So the number is 3,00×10m, wherem= 8, because there are 8 digits left after the decimal point. So, the speed of light in scientic notation to two decimal places is3,00×108 ms−1.
As another example, the size of the HI virus is around1,2×10−7m. This is equal to1,2×0,0000001m, which is 0,00000012 m.
Real Numbers
Now that we have learnt about the basics of mathematics, we can look at what real numbers are in a little more detail. The following are examples of real numbers and it is seen that each number is written in a dierent way.
√
3, 1,2557878, 56
34, 10, 2,1, −5, −6,35, − 1
90 (33)
Depending on how the real number is written, it can be further labelled as either rational, irrational, integer or natural. A set diagram of the dierent number types is shown in Figure 3.
Q R Z N
Figure 3: Set diagram of all the real numbersR, the rational numbersQ, the integersZand the natural numbers N. The irrational numbers are the numbers not inside the set of rational numbers. All of the integers are also rational numbers, but not all rational numbers are integers.
Non-Real Numbers
All numbers that are not real numbers have imaginary components. We will not see imaginary numbers in this book but they come from√
−1. Since we won't be looking at numbers which are not real, if you see a number you can be sure it is a real one.
Natural Numbers
The rst type of numbers that are learnt about are the numbers that are used for counting. These numbers are called natural numbers and are the simplest numbers in mathematics:
0,1,2,3,4, ... (34)
Mathematicians use the symbolN0to mean the set of all natural numbers. These are also sometimes called whole numbers. The natural numbers are a subset of the real numbers since every natural number is also a real number.
Integers
The integers are all of the natural numbers and their negatives:
...−4,−3,−2,−1,0,1,2,3,4... (35) Mathematicians use the symbol Z to mean the set of all integers. The integers are a subset of the real numbers, since every integer is a real number.
Rational Numbers
The natural numbers and the integers are only able to describe quantities that are whole or complete. For example, you can have 4 apples, but what happens when you divide one apple into 4 equal pieces and share it among your friends? Then it is not a whole apple anymore and a dierent type of number is needed to describe the apples. This type of number is known as a rational number.
A rational number is any number which can be written as:
a
b (36)
whereaandb are integers andb6= 0.
The following are examples of rational numbers:
20 9 , −1
2 , 20 10, 3
15 (37)
Notation Tip
Rational numbers are any number that can be expressed in the form ab;a, b∈Z;b6= 0which means the set of numbers ab whenaandb are integers.
Mathematicians use the symbol Qto mean the set of all rational numbers. The set of rational numbers contains all numbers which can be written as terminating or repeating decimals.
Rational Numbers
All integers are rational numbers with a denominator of 1.
You can add and multiply rational numbers and still get a rational number at the end, which is very useful. If we have 4 integersa, b, candd, then the rules for adding and multiplying rational numbers are
a
b +dc = ad+bcbd
a
b ×dc = acbd (38)
Notation Tip
The statement "4 integers a, b, c and d" can be written formally as {a, b, c, d} ∈Z because the ∈ symbol means in and we say thata, b, canddare in the set of integers.
Two rational numbers (ab and dc) represent the same number ifad=bc. It is always best to simplify any rational number, so that the denominator is as small as possible. This can be achieved by dividing both the numerator and the denominator by the same integer. For example, the rational number 1000/10000can be divided by 1000 on the top and the bottom, which gives1/10. 23 of a pizza is the same as 128 (Figure 4).
Figure 4: 128 of the pizza is the same as 23 of the pizza.
You can also add rational numbers together by nding the lowest common denominator and then adding the numerators. Finding a lowest common denominator means nding the lowest number that both denomi- nators are a factor13of. A factor of a number is an integer which evenly divides that number without leaving a remainder. The following numbers all have a factor of 3
3,6,9,12,15,18,21,24, ... (39)
and the following all have factors of 4
4,8,12,16,20,24,28, ... (40)
The common denominators between 3 and 4 are all the numbers that appear in both of these lists, like 12 and 24. The lowest common denominator of 3 and 4 is the smallest number that has both 3 and 4 as factors, which is 12.
For example, if we wish to add 34+23, we rst need to write both fractions so that their denominators are the same by nding the lowest common denominator, which we know is 12. We can do this by multiplying 34 by 33 and 23 by 44. 33 and 44 are really just complicated ways of writing 1. Multiplying a number by 1 doesn't change the number.
3
4+23 = 34×33+23×44
= 3×34×3+2×43×4
= 129 +128
= 9+812
= 1712
(41)
13Some people say divisor instead of factor.
Dividing by a rational number is the same as multiplying by its reciprocal, as long as neither the numerator nor the denominator is zero:
a b ÷ c
d =a b.d
c =ad
bc (42)
A rational number may be a proper or improper fraction.
Proper fractions have a numerator that is smaller than the denominator. For example,
−1 2 , 3
15, −5
−20 (43)
are proper fractions.
Improper fractions have a numerator that is larger than the denominator. For example,
−10 2 ,15
13,−53
−20 (44)
are improper fractions. Improper fractions can always be written as the sum of an integer and a proper fraction.
Converting Rationals into Decimal Numbers Converting rationals into decimal numbers is very easy.
If you use a calculator, you can simply divide the numerator by the denominator.
If you do not have a calculator, then you have to use long division.
Since long division was rst taught in primary school, it will not be discussed here. If you have trouble with long division, then please ask your friends or your teacher to explain it to you.
Irrational Numbers
An irrational number is any real number that is not a rational number. When expressed as decimals, these numbers can never be fully written out as they have an innite number of decimal places which never fall into a repeating pattern. For example,√
2 = 1,41421356..., π = 3,14159265.... π is a Greek letter and is pronounced pie.
Real Numbers
1. Identify the number type (rational, irrational, real, integer) of each of the following numbers:
a. cd ifcis an integer and ifdis irrational.
b. 32 c. -25 d. 1,525
e. √ 10
Click here for the solution14
2. Is the following pair of numbers real and rational or real and irrational? Explain. √
4;18 Click here for the solution15
14See the le at <http://cnx.org/content/m31330/latest/http://www.fhsst.org/l3b>
15See the le at <http://cnx.org/content/m31330/latest/http://www.fhsst.org/l3j>
Mathematical Symbols
The following is a table of the meanings of some mathematical signs and symbols that you should have come across in earlier grades.
Sign or Symbol Meaning
> greater than
< less than
≥ greater than or equal to
≤ less than or equal to
Table 6
So if we write x > 5, we say that x is greater than 5 and if we write x ≥y, we mean that x can be greater than or equal toy. Similarly, < means `is less than' and≤means `is less than or equal to'. Instead of saying that x is between 6 and 10, we often write 6 < x < 10. This directly means `six is less than x which in turn is less than ten'.
Mathematical Symbols
1. Write the following in symbols:
a. xis greater than 1
b. y is less than or equal to z c. ais greater than or equal to 21
d. pis greater than or equal to 21 andpis less than or equal to 25 Click here for the solution16
Innity
Innity (symbol ∞) is usually thought of as something like the largest possible number" or the furthest possible distance". In mathematics, innity is often treated as if it were a number, but it is clearly a very dierent type of number" than integers or reals.
When talking about recurring decimals and irrational numbers, the term innite was used to describe never-ending digits.
End of Chapter Exercises
1. Calculate a. 18−6×2 b. 10 + 3 (2 + 6)
c. 50−10 (4−2) + 6 d. 2×9−3 (6−1) + 1
e. 8 + 24÷4×2 f. 30−3×4 + 2 g. 36÷4 (5−2) + 6 h. 20−4×2 + 3
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i. 4 + 6 (8 + 2)−3 j. 100−10 (2 + 3) + 4 Click here for the solution17
2. Ifp=q+ 4r, thenr=... Click here for the solution18 3. Solve x−23 =x−3 Click here for the solution19
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Rational Numbers
Introduction
As described in the chapter on review of past work, a number is a way of representing quantity. The numbers that will be used in high school are all real numbers, but there are many dierent ways of writing any single real number.
This chapter describes rational numbers.
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Figure 1
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The Big Picture of Numbers
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The term whole number does not have a consistent denition. Various authors use it in many dierent ways.
We use the following denitions:
• natural numbers are (1, 2, 3, ...)
• whole numbers are (0, 1, 2, 3, ...)
• integers are (... -3, -2, -1, 0, 1, 2, 3, ....)
Denition
The following numbers are all rational numbers.
10 1 , 21
7 , −1
−3, 10 20, −3
6 (1)
You can see that all denominators and all numerators are integers.
Denition 1: Rational Number
A rational number is any number which can be written as:
a
b (2)
whereaandbare integers and b6= 0.
tip: Only fractions which have a numerator and a denominator (that is not 0) that are integers are rational numbers.
This means that all integers are rational numbers, because they can be written with a denominator of 1.
Therefore
√2 7 , π
20 (3)
are not examples of rational numbers, because in each case, either the numerator or the denominator is not an integer.
A number may not be written as an integer divided by another integer, but may still be a rational number. This is because the results may be expressed as an integer divided by an integer. The rule is, if a number can be written as a fraction of integers, it is rational even if it can also be written in another way as well. Here are two examples that might not look like rational numbers at rst glance but are because there are equivalent forms that are expressed as an integer divided by another integer:
−1,33
−3 =133 300, −3
6,39 =−300
639 =−100213 (4)
Rational Numbers
1. If a is an integer, b is an integer and c is irrational, which of the following are rational num- bers?
(i) 56 (ii) a3 (iii) b2 (iv) 1c Table 1
Click here for the solution21
21See the le at <http://cnx.org/content/m31331/latest/http://www.fhsst.org/l35>
2. If a1 is a rational number, which of the following are valid values fora? (i) 1 (ii)−10 (iii)√
2 (iv)2,1 Table 2
Click here for the solution22
Forms of Rational Numbers
All integers and fractions with integer numerators and denominators are rational numbers. There are two more forms of rational numbers.
Investigation : Decimal Numbers
You can write the rational number 12 as the decimal number 0,5. Write the following numbers as decimals:
1. 14 2. 101 3. 25 4. 1001 5. 23
Do the numbers after the decimal comma end or do they continue? If they continue, is there a repeating pattern to the numbers?
You can write a rational number as a decimal number. Two types of decimal numbers can be written as rational numbers:
1. decimal numbers that end or terminate, for example the fraction 104 can be written as 0,4.
2. decimal numbers that have a repeating pattern of numbers, for example the fraction 13 can be written as 0,˙3. The dot represents recurring 3's i.e.,0,333...= 0,˙3.
For example, the rational number 56 can be written in decimal notation as0,8 ˙3 and similarly, the decimal number 0,25 can be written as a rational number as 14.
tip: You can use a bar over the repeated numbers to indicate that the decimal is a repeating decimal.
Converting Terminating Decimals into Rational Numbers
A decimal number has an integer part and a fractional part. For example10,589 has an integer part of 10 and a fractional part of0,589because10 + 0,589 = 10,589. The fractional part can be written as a rational number, i.e. with a numerator and a denominator that are integers.
Each digit after the decimal point is a fraction with a denominator in increasing powers of ten. For example:
• 101 is0,1
• 1001 is0,01
22See the le at <http://cnx.org/content/m31331/latest/http://www.fhsst.org/l3N>
This means that:
10,589 = 10 +105 +1008 +10009
= 101000589
= 105891000
(5)
Fractions
1. Write the following as fractions:
(a) 0,1 (b)0,12 (c)0,58 (d)0,2589 Table 3
Click here for the solution23
Converting Repeating Decimals into Rational Numbers
When the decimal is a repeating decimal, a bit more work is needed to write the fractional part of the decimal number as a fraction. We will explain by means of an example.
If we wish to write0,˙3in the form ab (whereaandbare integers) then we would proceed as follows
x = 0,33333...
10x = 3,33333... multiply by 10 on both sides
9x = 3 (subtracting the second equation from the rst equation)
x = 39 =13
(6)
And another example would be to write5,˙4 ˙3 ˙2as a rational fraction.
x = 5,432432432...
1000x = 5432,432432432... multiply by 1000 on both sides
999x = 5427 (subtracting the second equation from the rst equation)
x = 5427999 = 20137
(7)
For the rst example, the decimal was multiplied by 10 and for the second example, the decimal was multiplied by 1000. This is because for the rst example there was only one digit (i.e. 3) recurring, while for the second example there were three digits (i.e. 432) recurring.
In general, if you have one digit recurring, then multiply by 10. If you have two digits recurring, then multiply by 100. If you have three digits recurring, then multiply by 1000. Can you spot the pattern yet?
The number of zeros is the same as the number of recurring digits.
Not all decimal numbers can be written as rational numbers. Why? Irrational decimal numbers like
√2 = 1,4142135...cannot be written with an integer numerator and denominator, because they do not have a pattern of recurring digits. However, when possible, you should try to use rational numbers or fractions instead of decimals.
23See the le at <http://cnx.org/content/m31331/latest/http://www.fhsst.org/l3R>
Repeated Decimal Notation
1. Write the following using the repeated decimal notation:
a. 0,11111111...
b. 0,1212121212...
c. 0,123123123123...
d. 0,11414541454145...
Click here for the solution24
2. Write the following in decimal form, using the repeated decimal notation:
a. 23 b. 1113
c. 456 d. 219
Click here for the solution25
3. Write the following decimals in fractional form:
a. 0,633 ˙3 b. 5,313131
c. 0,99999 ˙9
Click here for the solution26
Summary
1. Real numbers can be either rational or irrational.
2. A rational number is any number which can be written as ab where aandbare integers andb6= 0 3. The following are rational numbers:
a. Fractions with both denominator and numerator as integers.
b. Integers.
c. Decimal numbers that end.
d. Decimal numbers that repeat.
End of Chapter Exercises
1. Ifais an integer, bis an integer andc is irrational, which of the following are rational numbers?
a. 56 b. a3 c. b2 d. 1c
Click here for the solution27
2. Write each decimal as a simple fraction:
a. 0,5 b. 0,12
c. 0,6 d. 1,59
e. 12,27 ˙7
24See the le at <http://cnx.org/content/m31331/latest/http://www.fhsst.org/l3U>
25See the le at <http://cnx.org/content/m31331/latest/http://www.fhsst.org/l3n>
26See the le at <http://cnx.org/content/m31331/latest/http://www.fhsst.org/l3Q>
27See the le at <http://cnx.org/content/m31331/latest/http://www.fhsst.org/l3v>
Click here for the solution28
3. Show that the decimal 3,21 ˙1 ˙8is a rational number.
Click here for the solution29
4. Express0,7 ˙8as a fraction ab wherea, b∈Z(show all working).
Click here for the solution30
28See the le at <http://cnx.org/content/m31331/latest/http://www.fhsst.org/l3f>
29See the le at <http://cnx.org/content/m31331/latest/http://www.fhsst.org/l3G>
30See the le at <http://cnx.org/content/m31331/latest/http://www.fhsst.org/lOf>
Exponentials
Introduction
In this chapter, you will learn about the short cuts to writing2×2×2×2 . This is known as writing a number in exponential notation.
Denition
Exponential notation is a short way of writing the same number multiplied by itself many times. For example, instead of5×5×5, we write53 to show that the number 5 is multiplied by itself 3 times and we say 5 to the power of 3. Likewise52 is5×5 and35 is3×3×3×3×3. We will now have a closer look at writing numbers using exponential notation.
Denition 1: Exponential Notation
Exponential notation means a number written like
an (1)
where n is an integer and a can be any real number. a is called the base and n is called the exponent or index.
Thenth power ofais dened as:
an =a×a× · · · ×a (n times) (2) withamultiplied by itselfntimes.
We can also dene what it means if we have a negative exponent −n. Then,
a−n= 1
a×a× · · · ×a (ntimes) (3) tip: Exponentials
Ifnis an even integer, thenanwill always be positive for any non-zero real numbera. For example, although
−2is negative,(−2)2=−2× −2 = 4is positive and so is(−2)−2= −2×−21 = 14.
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Khan Academy video on Exponents - 1
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Figure 1
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Figure 2
Laws of Exponents
There are several laws we can use to make working with exponential numbers easier. Some of these laws might have been seen in earlier grades, but we will list all the laws here for easy reference and explain each law in detail, so that you can understand them and not only remember them.
a0 = 1 am×an = am+n
a−n = a1n am÷an = am−n
(ab)n = anbn (am)n = amn
(4)
Exponential Law 1: a
0= 1
Our denition of exponential notation shows that
a0 = 1 , (a6= 0) (5)
For example,x0= 1and(1 000 000)0= 1.
Application using Exponential Law 1: a0= 1,(a6= 0) 1. 160
2. 16a0 3. (16 +a)0 4. (−16)0
5. −160
Click here for the solution32
Exponential Law 2: a
m× a
n= a
m+nKhan Academy video on Exponents - 3
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Figure 3
Our denition of exponential notation shows that
am×an = 1×a×...×a (mtimes)
×1×a×...×a (ntimes)
= 1×a×...×a (m+ntimes)
= am+n
(6)
For example,
27×23 = (2×2×2×2×2×2×2)×(2×2×2)
= 27+3
= 210
(7)
note: This simple law is the reason why exponentials were originally invented. In the days before calculators, all multiplication had to be done by hand with a pencil and a pad of paper.
Multiplication takes a very long time to do and is very tedious. Adding numbers however, is very easy and quick to do. If you look at what this law is saying you will realise that it means that adding the exponents of two exponential numbers (of the same base) is the same as multiplying the two numbers together. This meant that for certain numbers, there was no need to actually multiply the numbers together in order to nd out what their multiple was. This saved mathematicians a lot of time, which they could use to do something more productive.
Application using Exponential Law 2: am×an=am+n 1. x2·x5
2. 23.24 [Take note that the base (2) stays the same.]
3. 3×32a×32
Click here for the solution33
32See the le at <http://cnx.org/content/m31332/latest/http://www.fhsst.org/lOG>
33See the le at <http://cnx.org/content/m31332/latest/http://www.fhsst.org/lO7>
Exponential Law 3: a
−n=
a1n, a 6= 0
Our denition of exponential notation for a negative exponent shows that a−n = 1÷a÷...÷a (ntimes)
= 1×a×···×a1 (ntimes)
= a1n
(8)
This means that a minus sign in the exponent is just another way of showing that the whole exponential number is to be divided instead of multiplied.
For example,
2−7 = 2×2×2×2×2×2×21
= 217
(9)
Application using Exponential Law 3: a−n= a1n, a6= 0 1. 2−2= 212
2. 23−22
3. 23−3
4. nm−4
5. aa−35·x·x−24
Click here for the solution34
Exponential Law 4: a
m÷ a
n= a
m−nWe already realised with law 3 that a minus sign is another way of saying that the exponential number is to be divided instead of multiplied. Law 4 is just a more general way of saying the same thing. We get this law by multiplying law 3 byam on both sides and using law 2.
am
an = ama−n
= am−n (10)
For example,
27÷23 = 2×2×2×2×2×2×2 2×2×2
= 2×2×2×2
= 24
= 27−3
(11)
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Figure 4
Application using Exponential Law 4: am÷an=am−n 1. aa62 =a6−2
2. 3326
3. 32a4a82
4. aa3x4
Click here for the solution35
Exponential Law 5: (ab)
n= a
nb
nThe order in which two real numbers are multiplied together does not matter. Therefore, (ab)n = a×b×a×b×...×a×b (ntimes)
= a×a×...×a (ntimes)
×b×b×...×b (ntimes)
= anbn
(12)
For example,
(2·3)4 = (2·3)×(2·3)×(2·3)×(2·3)
= (2×2×2×2)×(3×3×3×3)
= 24
× 34
= 2434
(13)
Application using Exponential Law 5: (ab)n=anbn 1. (2xy)3= 23x3y3
2. 7ab2 3. (5a)3
Click here for the solution36
35See the le at <http://cnx.org/content/m31332/latest/http://www.fhsst.org/lOA>
36See the le at <http://cnx.org/content/m31332/latest/http://www.fhsst.org/lOs>
Exponential Law 6: (a
m)
n= a
mnWe can nd the exponential of an exponential of a number. An exponential of a number is just a real number. So, even though the sentence sounds complicated, it is just saying that you can nd the exponential of a number and then take the exponential of that number. You just take the exponential twice, using the answer of the rst exponential as the argument for the second one.
(am)n = am×am×...×am (ntimes)
= a×a×...×a (m×ntimes)
= amn
(14)
For example,
223
= 22
× 22
× 22
= (2×2)×(2×2)×(2×2)
= 26
= 2(2×3)
(15)
Application using Exponential Law 6: (am)n=amn 1. x34
2. h a43i2
3. 3n+32
Click here for the solution37 Exercise 1: Simplifying indices
Simplify: 52x−1152x−3·9x−2
Investigation : Exponential Numbers
Match the answers to the questions, by lling in the correct answer into the Answer column. Possible answers are: 32, 1,−1,−13, 8. Answers may be repeated.
Question Answer 23
73−3
2 3
−1
87−6 (−3)−1 (−1)23
Table 1
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The following video gives an example on using some of the concepts covered in this chapter.
Khan Academy video on Exponents - 5
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Figure 5
Summary
Exponential notation means a number written like an where n is an integer and a can be any real number.
ais called the base andnis called the exponent or index.
Thenth power ofais dened as: an=a×a× · · · ×a (n times)
There are six laws of exponents:
· Exponential Law 1: a0= 1
· Exponential Law 2: am×an=am+n
· Exponential Law 3: a−n=a1n, a6= 0
· Exponential Law 4: am÷an=am−n
· Exponential Law 5: (ab)n =anbn
· Exponential Law 6: (am)n=amn
End of Chapter Exercises
1. Simplify as far as possible:
a. 3020 b. 10
c. (xyz)0 d. h
3x4y7z125
−5x9y3z42i0 e. (2x)3
f. (−2x)3 g. (2x)4 h. (−2x)4
Click here for the solution38
2. Simplify without using a calculator. Leave your answers with positive exponents.
a. (3x)3x−32
b. 5x0+ 8−2− 12−2
·1x c. 55b−3b+1
Click here for the solution39 3. Simplify, showing all steps:
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39See the le at <http://cnx.org/content/m31332/latest/http://www.fhsst.org/lOu>
a. 2a−26.3aa+3
b. aam+n+p2m+n+p·am
c. 327n·9n−1n−3
d.
2x2a y−b
3
e. 23x−142x−2·8x+1 f. 2262x−12x·11·32x2x
Click here for the solution40 4. Simplify, without using a calculator:
a. (−3)(−3)−3·(−3)−4 2
b. 3−1+ 2−1−1
c. 9n−181·272−n3−2n d. 23n+243n−2·8n−3
Click here for the solution41
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41See the le at <http://cnx.org/content/m31332/latest/http://www.fhsst.org/lOh>
Estimating Surds
Introduction
You should know by now what the nth root of a number means. If the nth root of a number cannot be simplied to a rational number, we call it asurd. For example, √
2 and √3
6 are surds, but√
4is not a surd because it can be simplied to the rational number 2.
In this chapter we will only look at surds that look like √n
a, whereais any positive number, for example
√7 or √3
5. It is very common forn to be 2, so we usually do not write √2
a. Instead we write the surd as just√
a, which is much easier to read.
It is sometimes useful to know the approximate value of a surd without having to use a calculator. For example, we want to be able to estimate where a surd like√
3 is on the number line. So how do we know where surds lie on the number line? From a calculator we know that√
3 is equal to1,73205.... It is easy to see that√
3is above 1 and below 2. But to see this for other surds like√
18without using a calculator, you must rst understand the following fact:
note: Ifa andb are positive whole numbers, and a < b, then √n a < √n
b. (Challenge: Can you explain why?)
If you don't believe this fact, check it for a few numbers to convince yourself it is true.
How do we use this fact to help us guess what√
18is? Well, you can easily see that 18<25. Using our rule, we also know that√
18<√
25. But we know that52= 25so that√
25 = 5. Now it is easy to simplify to get√
18<5. Now we have a better idea of what√ 18is.
Now we know that √
18is less than 5, but this is only half the story. We can use the same trick again, but this time with 18 on the right-hand side. You will agree that 16<18. Using our rule again, we also know that√
16<√
18. But we know that 16 is a perfect square, so we can simplify√
16to 4, and so we get 4<√
18!
As you can see, we have shown that√
18is between 4 and 5. If we check on our calculator, we can see that √
18 = 4,1231..., and the idea was right! You will notice that our idea used perfect squares that were close to the number 18. We found the largest perfect square smaller than 18 was42= 16, and the smallest perfect square greater than 18 was52= 25. Here is a quick summary of what a perfect square or cube is:
note: A perfect square is the number obtained when an integer is squared. For example, 9 is a perfect square since32= 9. Similarly, a perfect cube is a number which is the cube of an integer.
For example, 27 is a perfect cube, because33= 27.
To make it easier to use our idea, we will create a list of some of the perfect squares and perfect cubes. The list is shown in Table 1.
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